A. Background
The Kirchhoff integral formulation is one of the most widely used methods for predicting acoustic radiation and scattering from an elastic structure in engineering practice. The advantage of using this integral formulation is a reduction of the dimensionality of the problem by one. The first step of this approach is to determine the acoustic quantities on the surface. For an acoustic radiation problem, the normal component of the particle velocity (or the surface acoustic pressure for an acoustic scattering problem) must be specified. Since the surface is impermeable, the normal component of the particle velocity is equal to that of the surface velocity, which can be measured by an accelerometer. Next, the surface acoustic pressure (or the normal component of the particle velocity for acoustic scattering) is determined by solving an integral equation. Once these quantities are known, the radiated acoustic pressure anywhere can be calculated by the Kirchhoff integral formulation.
It may be difficult to use a conventional accelerometer to measure the vibration response of a flexible or a lightweight structure such as a loudspeaker membrane or a passenger vehicle fuel pump, because the weight of the accelerometer may alter the desired signal. In other cases, it may be unfeasible to use an accelerometer on a structure with a hostile environment such as an engine oil pan, where the temperature on the surface is extremely high. Under these circumstances, we must rely on a non-intrusive measurement technique.
One approach commonly adopted in practice is to use a laser vibrometer to measure the normal component of the particle velocity, which is equal to that of the surface velocity at the interface, and then solve an integral equation for the surface acoustic pressure. The shortcomings of such an approach are well-known: (1) the surface Kirchhoff integral equation may fail to yield a unique solution whenever the frequency is close to one of the eigenfrequencies associated with to the related interior boundary value problem, and (2) the numerical computation may become quite involved. This is because for an arbitrary surface, we must discretize the surface into many segments with several hundreds or even more nodes. Accordingly, we must solve a large number of simultaneous integral equations for the acoustic pressures at these nodes using boundary element method (BEM). Since the central processing unit (CPU) time increases quadratically with the number of the nodes, the computation process may be excessively time-consuming.
Actually, the laser technique can be used to measure the displacement and velocity vectors of a suspended microparticle in an insonified medium. The work in this area, however, has received much less attention than that of measurements of the out-of-plane motion of a vibrating structure. Summarized below are the basic principles and applications of the non-intrusive laser measurement techniques to measurements of the particle displacement and velocity vectors both in fluids and in air.
B. Laser techniques
1. Laser Doppler velocimeter (LDV)
LDV has become a standard tool for non-intrusive measurements of fluid particle velocities. The basic premise in the LDV measurements is that motion of the microparticles in the fluid (either due to natural impurities or due to seeded particles) will scatter the incident light, and produce a Doppler shift in the scattered light which can be detected with appropriate electronics and signal processing. LDV used in the fluid mechanics was extended to the acoustics by measuring the in-air particle velocities associated with steady-state time-harmonic standing waves and travelling waves inside a tube. Laser Doppler anemometry (LDA) has been used for the remote detection of sound. The technique of LDA consists of measurements of the velocity of neutrally buoyant microparticles suspended in an acoustic field by analyzing the spectral content of Doppler-shifted laser light scattered by the microparticles. LDV has been used to measure the acoustic particle velocity in fluids. In particular, the measurements of acoustic particle displacements using different LDV systems has determined that LDV was capable of detecting the particle displacements in the order of a few nanometers with a bandwidth of several kilohertz. The performance and limitations of LDV systems were also analyzed, and the effect of Brownian motion (i.e., thermal agitation in the fluid) on the measured data was shown to produce only negligible broadening of the spectral density of the signal of interest. An equation of motion of microparticles in suspension in an insonified fluid has been derived and it has been determined that the motion of neutrally buoyant microparticles closely emulates the displacement of the surrounding insonified fluid and confirms the basic tenet associated with the laser detection of sound.
2. Differential laser Doppler interferometry (DLDI)
DLDI is evolved from the principle of LDA and used to measure simultaneously the out-of-plane and the in-plane velocities on the surface of a vibrating object. The principle of DLDI is to measure the phase shift of the reflected or scattered light from the surface due to surface vibrational motion. The main component of a DLDI system is a probe head that has three illuminating single mode fibers. Prior to launching, the laser beams are frequency shifted by three acousto-optic Bragg cells by 40.0, 40.1, and 40.3 MHz, respectively, so the interference between the first and second beams occurs at 100 kHz, while those between the second and third and the first and third occur at 200 and 300 kHz, respectively. Geometrically, the first and second beams are positioned symmetrically with respect to the unit normal on the surface at an angle .alpha., and the third beam is aligned with the first and second beams at an angle .beta.(.beta.&lt;.alpha.) with respect to the unit normal. In the differential configuration, the 100 kHz carrier will be modulated by the in-plane motion, and the 200 and 300 kHz carriers will be modulated by both in-plane and out-of-plane motions, respectively.
Mathematically, the surface displacement vector can be written as EQU x.sub.S (t)=u.sub.in (t)e.sub.in +u.sub.out (t)e.sub.out ( 1)
where u.sub.in (t) and u.sub.out (t) represent the in-place and out-of-place components of the surface displacement, respectively, and e.sub.in and e.sub.out are the unit vectors in the corresponding directions.
Accordingly, the phase terms .phi..sub.ij, where i, j=1 to 3, in the 100, 200, and 300 kHz carriers can be written as EQU .phi..sub.12 =2ku.sub.in (t) sin .alpha. (2a) EQU .phi..sub.13 =(sin .alpha.+sin .beta.)ku.sub.in +(cos .beta.-cos .alpha.)ku.sub.out (t) (2b) EQU .phi..sub.23 =(sin .alpha.+sin .beta.)ku.sub.in -(cos .beta.-cos .alpha.)ku.sub.out (t) (2c)
where k is the optical wavenumber. Hence by measuring the phase shifts .phi..sub.ij in the 100, 200, and 300 kHz carriers, one can determine the displacement vector on the surface. Since there are only two unknowns, one can use any two equations, say, Eqs. (2a) and (2b) to specify u.sub.in and u.sub.out.
In one optical system, the demodulation is done by using a combination of filters and the phase-locked loops (PLL). The PLL demodulates the signal with phase .phi..sub.ij and generates an output which is proportional to the time rate of changes of .phi..sub.ij. From Eqs. (2a) and (2b), one finds ##EQU1##
Therefore, by measuring the instantaneous frequency deviations d.phi..sub.12 /dt and d.phi..sub.13 /dt from the carrier frequencies at 100 and 300 kHz, one can determine simultaneously the in-plane and out-of-plane components of the surface velocity. A three dimensional laser vibrometer was designed based on this principle to measure simultaneously the three components of the velocity on the surface of a vibrating structure.
The DLDI technique can be extended in principle to the measurement of the velocity of a microparticle in the vicinity of a vibrating object. Imagine that an object is surrounded by neutrally buoyant microparticles. As the object vibrates, the acoustic pressure fluctuations will excite the microparticles into oscillations. Suppose that we define a control surface and focus the laser beams on a microparticle lying on that surface. There is no restriction on the formation of the control surface so long as it completely encloses the vibrating object. In the special case in which the control surface coincides with the vibrating surface, the normal component of the displacement of the microparticle will be equal to that of the surface displacement, while the tangible components may be different. In any event, the microparticle displacement in the directions normal and tangential to the control surface will cause a Doppler shift in the phase .phi..sub.ij of the reflected light, which is modulated in the frequency carriers. Once the signal with phase .phi..sub.ij is demodulated, we can calculate the microparticle velocity which is proportional to the time derivative of the phase, d.phi..sub.ij /dt.
3. Electronic speckle pattern interferometry (ESPI)
Alternatively, we can use ESPI to measure the phase term of a microparticle, which is an established optical technique for measuring static and dynamic deformations and surface shapes for more than two decades. Specifically, we can utilize the stroboscopic technique, which "freezes" the dynamic motion of a particle at one position so that during other times of the movement cycle, the particle is not illuminated and therefore is "invisible" to the imaging device. In practice, this technique can be implemented by using a pulsed laser or a light shuttering device with a continuous wave laser. The time interval between two consecutive pulses or shutters is typically in the range of nanoseconds, so ESPI can capture very high frequency oscillations. By using an additive-subtractive speckle pattern interferometry, the accuracy of the phase measurement can be further enhanced.
Suppose that we take five frames of additive speckle patterns of the motion of a microparticle in suspension S.sub.j, j=0, 1 . . . , and 4. Here the speckle pattern S.sub.o is taken with the laser illumination pulsed at the instant when the microparticle reaches its zero amplitude of a harmonic oscillation. The remaining four speckle patterns, S.sub.1 to S.sub.4, are taken with the laser illumination pulsed at the instants when the microparticle reaches its maximum and minimum amplitudes. During the acquisition of S.sub.0 to S.sub.4, the phase of the reference beam is shifted appropriately and is synchronized with the pulses. Accordingly, the displacement-induced phase term .phi. of a microparticle at any surface point x.sub.S can be written as ##EQU2## where F.sub.n, n=0, 1, 2, and 3, are the additive-subtractive fringe patterns which have the same form as that of the Michelson interferometric fringe pattern, except for the randomly distributed modulation term B/cos .psi./ contributed by the speckles, EQU F.sub.n =B.vertline.cos .psi..vertline.1-cos (.phi.+n.pi./2)!(5)
Once the phase term .phi. is determined, the microparticle velocity which is proportional to the time derivative of the phase can be specified.